Individual score contributions in ML estimation

Question

I've estimated a model via maximum likelihood in Stata and was surprised to find that estimated standard errors for one particular parameter are drastically smaller when clustering observations. I take it from the Stata manual on robust standard error estimation in ML that this can happen if the contributions of individual observations to the score (the derivative of the log-likelihood) tend to cancel each other within clusters.

I would now like to dig a little deeper into what exactly is happening and would therefore like to have a look at these score contributions. As far as I can see, however, Stata only gives me the total sum as e(gradient). Is there any way to pry the individual summands out of Stata?

Answer 1

If you have written your own command, you can create a new variable containing these scores using the ml score command. Official Stata commands and most finished user written commands will often have score as an option for predict, which does the same thing but with an easier syntax.

These will give you the score of the log likelihood ($\ell$) with respect to the linear predictor, $x\beta = \beta_0 + \beta_1 x_1 + \beta_2 x_2 \elipses$. To get the derivative of the log likelihood with respect to an individual parameter, say $\beta_1$, you just use the chain rule:

$\frac{\partial \ell}{\partial \beta_1} = \frac{\partial \ell }{\partial x\beta} \frac{\partial x\beta}{\partial \beta_1}$

The scores returned by Stata are $ \frac{\partial \ell }{\partial x\beta}$, and $\frac{\partial x\beta}{\partial \beta_1} = x_1$.

So, to get the score for $\beta_1$ you just multiply the score returned by Stata and $x_1$.